Decorrelation and efficient coding by retinal ganglion cells

Abstract

An influential theory of visual processing asserts that retinal center-surround receptive fields remove spatial correlations in the visual world, producing ganglion cell spike trains that are less redundant than the corresponding image pixels. For bright, high-contrast images, this decorrelation would enhance coding efficiency in optic nerve fibers of limited capacity. We tested the central prediction of the theory and found that the spike trains of retinal ganglion cells were indeed decorrelated compared with the visual input. However, most of the decorrelation was accomplished not by the receptive fields, but by nonlinear processing in the retina. We found that a steep response threshold enhanced efficient coding by noisy spike trains and that the effect of this nonlinearity was near optimal in both salamander and macaque retina. These results offer an explanation for the sparseness of retinal spike trains and highlight the importance of treating the full nonlinear character of neural codes.

Figure 1

Decorrelation of naturalistic stimuli. a, b: Sample frames of naturalistic and white noise stimuli, projected onto a 3.4mm square on the retina. c: Responses of two retinal ganglion cells to a short segment of the naturalistic stimulus, displayed as rasters of spikes on 250 identical repeats. d: A sample spatio-temporal receptive field for an OFF ganglion cell, measured as the spike-triggered average stimulus and integrated over one spatial dimension for ease of display. Note the spatial center-surround antagonism (red regions above and below blue) and the biphasic timecourse (red region left of blue). e: Spatial receptive fields of two OFF cells, including 1-s.d. outlines of the receptive field centers (solid) and surrounds (dotted). f: Cross-correlation function between two ganglion cell spike trains, indicating the frequency of spike pairs as a function of their delay. The shaded area encompasses most of the central peak and indicates the range of delays used to compute the quoted correlation coefficients. g, h: Correlation coefficient between the responses of two ganglion cells as a function of their distance under a white noise (g) or naturalistic (h) stimulus. Each pair of cells contributes a point; lines represent median correlation for pairs at similar distance. Comparisons are restricted within a cell type (solid lines) or across cell types (dashed lines). For reference, the correlation between stimulus pixels is shown as well (thin lines).

Figure 2

Nonlinearity accounts for much of decorrelation. a, b: Spatial correlation functions for neurons and models under naturalistic stimulation. Cells with the same polarity preference (OFF-OFF or ON-ON pairs) have positive correlations (a) and those with opposite polarity preferences (OFF-ON pairs) have negative correlations (b). Curves are displayed as in Fig. 1h for the stimulus, the trial-averaged firing rates, the spike trains, and linear models. In panel b the stimulus correlations are shown with opposite sign for ease of comparison. Results from many cell pairs are summarized by the median correlation for pairs at similar retinal distance; error bars indicate the central quartiles. c: The origins of decorrelation in different response components. The full circle represents the median correlation present in the stimulus after filtering by the receptive field centers, at a retinal distance of 300 μm (arrowheads in panel a). The empty wedge (C) is the much smaller remaining correlation between the ganglion cell spike trains. The red wedge represents the decorrelation caused by lateral inhibition from receptive field surrounds. The difference between the linear response and the observed firing rate is due to nonlinear processing, and is responsible for over half the decorrelation implemented by the retina (green wedge). The trial-to-trial variation contributes an additional small amount of decorrelation (blue wedge). d: Decorrelation in the time domain. Autocorrelation functions of salamander ganglion cell responses and linear models are plotted as a function of delay during naturalistic stimulation. The linear filter’s first lobe of ~100 ms width (inset, black) introduced excess correlation beyond that in the stimulus. The antagonistic second lobe (inset, red) counteracted those but overcompensated, introducing anticorrelations at long delays. The observed correlations in the firing rate are much smaller still. e, f: Spatial (e) and temporal (f) correlations in macaque retinal ganglion cells, displayed as in (a, b, d). Macaque RGC responses were approximated by an LN model ^,, using published spatio-temporal receptive field parameters (Eqns 4–6) and sigmoidal nonlinearities (Eqn 10). The output noise was modeled as sub-Poisson variation (Eqn 11) with parameters derived from published spike trains as described in Methods. The stimulus was scaled in space and time to compensate for the different scales of primate and salamander receptive fields. L: Receptive field filter only. LN: including the nonlinearity. LN+noise: including the noise.

Figure 3

Sparseness in retinal responses. a: Spike rasters for two salamander ganglion cells over 10 repetitions of a naturalistic stimulus. Firing events are brief, separated by long silences, and have some trial-to-trial variability. b: Mean firing rates for the same neurons, with shading that indicates the standard deviation about the mean in time bins of 50 ms. c: The linear response generated from convolving the stimulus with the spatiotemporal receptive fields of those two cells. This linear model generally captures the times of firing events but differs dramatically in sparseness. d: Depiction of three factors contributing to decorrelation between two caricatured neural responses: event timing, sparseness, and noise.

Figure 4

Decorrelation and efficient coding in the LNP model. a: Schematic of two visual neurons that each respond according to the linear-nonlinear-Poisson (LNP) model. For each cell (top and bottom), the stimulus is processed by a linear filter that includes lateral inhibition in space; this signal is passed through a sigmoid nonlinearity; the result modulates the rate of a Poisson process that generates spikes; the spike counts in discrete time windows are the response variable. b: Four sample nonlinearities with sigmoid shape and high or low gain (solid or dashed lines), high or low threshold (thick or thin lines), and various peak rates. The shaded curve indicates the probability distribution of the filtered stimulus signal at the input to the nonlinearity. c: The effects of such a nonlinear transform on the correlations between two jointly Gaussian variables (see text). Note that the output correlation is always less than that of the input. A low threshold (thin lines) affects the correlation only weakly, but at high threshold (thick lines) the output correlation is greatly reduced, especially for negative values. The precise shape of the nonlinearity (dashed vs solid) is less important, and the peak rate has no effect. d: The ratio of output correlation to input correlation decreases with increasing threshold, shown here for the sigmoid nonlinearity applied to two variables with input correlation C=±0.6. e: When the two outputs are affected by independent additive noise, this reduces the output correlation by a factor determined by the signal-to-noise ratio (Eqn 14). f: Influence of the nonlinearity on information transmission. Within the framework of the LNP model, the threshold and gain of the sigmoid nonlinearity determine how much information about the stimulus is transmitted by the spikes (grayscale and contour lines). The average firing rate was fixed at 1.1 Hz (the median over the salamander ganglion cells). Threshold and 1/gain are measured in standard deviations of the input signal distribution. Insets illustrate nonlinearities (solid lines) at different thresholds and gains relative to the input distribution (shaded area). g, h: When multiple neurons receive correlated inputs, raising the threshold makes their outputs more redundant (g) even as the total information increases (h) and correlation decreases (d). All neurons had pairwise correlation coefficients of 0.9, equal thresholds, optimal (infinite) gain, and a fixed mean firing rate of 1.1 Hz. The optimal threshold varies only weakly with population size (N=1,…,8).

Figure 5

Efficiency of stimulus coding by retinal ganglion cells. a: Cumulative distribution of the spike count in 50-ms time bins, averaged over multiple repeats of the stimulus. Data (thin lines) for three sample ganglion cells and their fit with a model (thick lines) parametrized by θ, g, and K (Eqn 20). b: The information transmitted by model firing rate distributions with a fixed mean firing rate of 1.1 Hz, whose shape is parametrized by θ and g. Noise was assumed to be sub-Poisson as observed empirically (Fig. S3, Eqn 11). The blue dot indicates the globally maximal rate of information transmission at this mean rate. Red dots indicate the parameters of the rate distribution measured from salamander ganglion cells. These cells have widely varying mean firing rates. The contour plot of information transmission varies slightly with mean rate, but is shown here for illustration purposes only at one typical mean rate. c: Histogram of information efficiencies over the population of salamander retinal ganglion cells. For each cell, the information rate is calculated directly from the empirical spike counts. To calculate efficiency, this information rate is compared to the maximal information rate possible for the measured mean firing rate (Methods). d: Information transmission estimated for macaque retinal ganglion cells, displayed as in (b). Red dots are parameters describing the firing rate distribution obtained from published spike rasters in response to white noise stimulation . The contour plot shows the information transmission for different firing rate distributions while fixing the mean rate and time window to typical values, namely 30 Hz and 10 ms respectively.